Coordinate geometry is a branch of geometry that defines the position of points on the plane using an ordered pair of numbers, commonly known as coordinates.
If a point p(x, y) divides the join of A(x_{1}, y_{1}) and B(x_{2}, y_{2}), in the ratio m: n, then
The equation of a line in slope intercept form is Y= mx+ c, where m is its slope.
The equation of a line which has gradient m and which passes through the point (x_{1}, y_{1}) is = y – y_{1} = m(x – x_{1}).
Q1: Given two points A(2, 5) and B(6, 9), find the Distance between points A(2, 5) and B(6, 9)
Ans: Distance between points A(2, 5) and B(6, 9):
Q2: If the slope of a line is 2/3 and it passes through the point (2, 5), find the equation of the line in pointslope form.
Ans: Equation of the line passing through points P(3, 8) and Q(5, 2):
Using the pointslope form: y – y_{1} = m(x – x_{1})
y – 8 = 5(x – 3)
y – 8 = 5x + 15
y = 5x + 23
Q3: A rectangle has vertices at points A(1, 1), B(5, 1), C(5, 3), and D(1, 3). Determine its area and perimeter.
Ans: Rectangle with vertices at points A(1, 1), B(5, 1), C(5, 3), and D(1, 3):
Area = Length * Width Length =
Distance between points B and C:
Length = √((5 – 5)^{2} + (3 – 1)^{2}) = √(0 + 4) = √4 = 2
Width = Distance between points A and B:
Width = √((5 – 1)^{2} + (1 – 1)^{2}) = √(16 + 0) = √16 = 4 Area = 2 * 4 = 8 square units
Q4: Determine the equation of the circle with center C(2, 1) and a radius of 5.
Ans: Equation of the circle with center C(2, 1) and radius 5:
The equation of a circle with center (h, k) and radius r is (x − h)^{2} + (y − k)2 = r^{2}
For this circle: = (x − h)^{2} + (y−(−1)^{2} = r^{2} = 5^{2} (x – 2)^{2} + (y + 1)^{2} = 25
Q5: Given the points A(3, 6), B(8, 6), and C(5, 2), find the area of the triangle ABC.
Ans: Area of triangle ABC with vertices A(3, 6), B(8, 6), and C(5, 2):
Area = 1/2 x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)
Area = 1/2 3(6 – 2) + 8(2 – 6) + 5(6 – 6)
Area = 1/2 12 – 24 + 0
Area = 1/2 12
Area = 6 square units
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